Dimensions are not same for the pair

Options

A. Power and Strain
B. Torque and Work
C. Momentum and Impulse
D. Pressure and Stress

Explanation

To solve the problem of identifying which pair of physical quantities has different dimensions, we will analyze each option provided in the question.

### Step-by-Step Solution:

**Step 1: Analyze Option A - Power and Strain**
- **Power (P)** is defined as work done per unit time:
\[ P = \frac{W}{t} \]
The dimension of work (W) is given by:
\[ [W] = [F \cdot d] = [M L^2 T^{-2}] \]
Therefore, the dimension of power is:
\[ [P] = \frac{[W]}{[t]} = \frac{[M L^2 T^{-2}]}{[T]} = [M L^2 T^{-3}] \]

- **Strain** is defined as the ratio of change in length to original length:
\[ \text{Strain} = \frac{\Delta L}{L} \]
Since both change in length and original length have the same dimension (length), strain is dimensionless:
\[ [\text{Strain}] = 1 \]

- **Conclusion for Option A**: The dimensions of power and strain are not the same.

**Step 2: Analyze Option B - Torque and Work**
- **Torque (τ)** is defined as force times the perpendicular distance:
\[ τ = F \cdot r \]
The dimension of torque is:
\[ [τ] = [F] \cdot [L] = [M L T^{-2}] \cdot [L] = [M L^2 T^{-2}] \]

- **Work (W)** has already been calculated as:
\[ [W] = [M L^2 T^{-2}] \]

- **Conclusion for Option B**: The dimensions of torque and work are the same.

**Step 3: Analyze Option C - Momentum and Impulse**
- **Momentum (p)** is defined as mass times velocity:
\[ p = m \cdot v \]
The dimension of momentum is:
\[ [p] = [M] \cdot [L T^{-1}] = [M L T^{-1}] \]

- **Impulse (I)** is defined as force times time:
\[ I = F \cdot t \]
The dimension of impulse is:
\[ [I] = [M L T^{-2}] \cdot [T] = [M L T^{-1}] \]

- **Conclusion for Option C**: The dimensions of momentum and impulse are the same.

**Step 4: Analyze Option D - Pressure and Stress**
- **Pressure (P)** is defined as force per unit area:
\[ P = \frac{F}{A} \]
The dimension of pressure is:
\[ [P] = \frac{[F]}{[L^2]} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

- **Stress** is also defined as force per unit area:
\[ \text{Stress} = \frac{F}{A} \]
Therefore, the dimension of stress is the same as that of pressure:
\[ [\text{Stress}] = [M L^{-1} T^{-2}] \]

- **Conclusion for Option D**: The dimensions of pressure and stress are the same.

### Final Conclusion:
From the analysis, we find that the only pair with different dimensions is **Option A: Power and Strain**.